My question is: 

> Are there an alternative proof of Cramer-Rao lower bound that does not use
> Cauchy-Swartz inequality?


Let me outline the classical proof and explain why I am interested in this question.

Choose some function $g(X,Y)$. Then,
\begin{align}
E[ (X-E[X|Y]) g(X,Y)] \le   \left|   E[ (X-E[X|Y]) g(X,Y)]  \right|  \le \sqrt{ E \left[ (X-E[X|Y])^2 \right] E[g(X,Y)^2] }.
\end{align}

Therefore,
\begin{align}
E \left[ (X-E[X|Y])^2 \right]  \ge  \frac{\left|   E[ (X-E[X|Y]) g(X,Y)]  \right|}{E[g(X,Y)^2]}.
\end{align}

The proof is completed by choosing $g(x,y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ and noting that then  $ E[ (X-E[X|Y]) g(X,Y)]=-1$. This gives us the Cramer-Rao lower bound
\begin{align}
E \left[ (X-E[X|Y])^2 \right]  \ge \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) )  \right)^2 \right]}.
\end{align}


The choice of $g(X,Y)=\frac{d}{dx} \log (f_{XY}(x,y) ) $ always seemed mysterious to me. That is why I am wondering whether there is a more "natural" proof where the quantity  $\frac{d}{dx} \log (f_{XY}(x,y) )$ appearance is more obvious. 

For example, can we find an identity 

\begin{align}
E \left[ (X-E[X|Y])^2 \right]  = \frac{1}{E \left[ \left(\frac{d}{dx} \log (f_{XY}(X,Y) )  \right)^2 \right]}+c. 
\end{align}
where $c$ is non-negative.